# How do you differentiate #f(x)=(x^3-4)(3x-3)# using the product rule?

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I divide the factors into f and g, find the derivative of each of them separately, and then apply the product rule to find f'(x).

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To differentiate ( f(x) = (x^3 - 4)(3x - 3) ) using the product rule:

- Identify the two functions being multiplied: ( u(x) = x^3 - 4 ) and ( v(x) = 3x - 3 ).
- Apply the product rule formula: ( (uv)' = u'v + uv' ).
- Differentiate each function separately:
- ( u'(x) ) is the derivative of ( x^3 - 4 ), which is ( 3x^2 ).
- ( v'(x) ) is the derivative of ( 3x - 3 ), which is ( 3 ).

- Substitute the derivatives and the original functions into the product rule formula:
- ( f'(x) = (3x^2)(3x - 3) + (x^3 - 4)(3) ).

- Simplify the expression:
- ( f'(x) = 9x^3 - 9x^2 + 3x^3 - 12 ).

- Combine like terms:
- ( f'(x) = 12x^3 - 9x^2 - 12 ).

So, the derivative of ( f(x) = (x^3 - 4)(3x - 3) ) using the product rule is ( f'(x) = 12x^3 - 9x^2 - 12 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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